A major must complete one of the major tracks A)- E) described below. For each track, the stated requirements assume that the sequence Calculus I-II-III or the equivalent has been completed and is noted on your permanent record.The choice of track depends partly on your plans after graduation, but also on your tastes and abilities. Here are some comments about each track.

Each student has the final responsibility to be sure the requirements for the major are fulfilled correctly. Be sure to read the rules about double counting, course substitutions, etc. that are listed below, following the comments on the tracks. If there is ever any confusion or doubt, please confer with your major advisor or the Associate Director of Undergraduate Studies, Dr. Blake Thornton

Most majors take all of the courses Math 309, 310, 318 (308) and 3200, so there is overlap between the tracks and a decision about which track to follow can usually be postponed until sometime in the sophomore year. Even after that, if courses have been planned carefully, a change of track may be possible. Of course, choosing a track as soon as it's practical will let you pick courses to develop the best possible program. If you want to discuss how the different tracks might fit into your plans, please contact Dr. Blake Thornton.

If you are interested in some computer science courses beyond the one required in some major tracks, then please read this advice on picking a CSE course. These comments were put together with the help of the Computer Science & Engineering Department.

Track A (Traditional) Total 27 upper level mathematics units

Math 3200 or Math 493Math 310Math 4111 and 4121Math 429 and 430Three other upper level mathematics courses (these could include Math 309, Math 318 (or 308)). At least one upper level elective must be at 400-level or higher.

CSE 131. This should be taken as soon as possible. It is a prerequisite for some 400-level prob/stats courses. Students with a sufficient background in computing may use Math 475 to fulfill the computer science requirement in this track. But in that case Math 475 cannot also count as one of the required upper level electives.Students who took CSE 200 (not longer offered by CSE Department) may count that course in place of CSE 131.

CSE 131. Students who took CSE 200 (not longer offered by CSE Department) may count that course in place of CSE 131. This requirement shoud be completed as soon as possible. It is one of the prerequisites for Math 449.Math 217 (a prerequisite for Math 308)

Track D (Math/Secondary Education) Only available to students who also major in secondary education. Total 24 upper level mathematics units

CSE 131. This should be taken as soon as possible. Students who took CSE 200 (not longer offered by CSE Department) may count that course in place of CSE 131Math 309Math 318 (prereq. Math 309), or Math 308 (prereq. Math 217)Math 310Math 3200 (or Math 493)Math 302Math 331Two other upper level electives

Economics 1011-1021One of Economics 4011, 4151, 467Math 309Math 3200 (or 493)Math 310Math 4111 and 4121Three additional upper level mathematics courses: one of these must be Math 429 or 493. The other two can be chosen from among Math 318 (or 308), Math 456, or any 400-level probability/statistics course.

The following items, however, apply to every track, so please read carefully

An "upper level" mathematics course is whose course number begins with a "3" or higher: L24 Math 3***. A course with a lower number does not count toward upper level mathematics requirements even if it is cross-listed as an upper level course in another department or program: for example, if Math 2200 were cross-listed by another department as Lxx-3xx, then registering for Lxx-3xx would not satisfy an upper-level mathematics requirement.

All required courses must be completed with a letter grade of C- or better: pass/fail not allowed. This applies also to any 100 or 200 level courses that are required in a major track. The only exception is for courses for which credit was awarded through AP, IB, or a similar program. If a student takes a required course for a “pass/fail” grade, perhaps before deciding to do a major or minor, then:

a) if we can verify from the course instructor that the grade in the course would have been a C- or better, then the student can choose another course at the same level or higher (300-level, 400-level, …) to replace the course taken P/F, or b) the student can retake the course for a letter grade. (In the latter case, the student might not be able to count the course twice (6 units) toward the required 120 units for the degree.)

Math 318 and 308 cannot both be used to fulfill major requirements

Math 308 (Mathematics for the Physical Sciences) may be a good choice for students interested in physics or engineering. The tools and examples from Math 308 are useful in those areas. Math 308 is offered only in the spring semester and it has both Math 233 and Math 217 as prerequisites.

For other majors, Math 318 is usually the better choice. It is offered in both the fall and spring semesters, and has both Math 233 and Math 309 as prerequisites. For many students planning to take Math 4111 and 4121, Math 318 is a good "bridge course" between calculus and the rigors of analysis. After consulting with your advisor, it may be possible for a stronger student to replace the requirement for Math 318/308 with Math 4111. In that case, if your track also requires Math 4111, then you would need to add another upper level elective in place of the 318/308 requirement so that the total number of upper level courses required in the major track is not reduced.

With departmental approval, courses transferred from other accredited colleges and universities can be counted toward the major. However, courses transferred from a 2-year college (such as a community college) cannot be used to satisfy upper level requirements. For example, if you took a community college course that seemed roughly similar to 309, we might transfer the credit "as" 309 but the course would not count toward the major. In such a case, you may ask the Mathematics Department for permission to let you to substitute an additional upper level elective in place of 309 rather than retaking the course. The College will not transfer courses taken online.

For transferred courses: at least half of the upper level credits required in a mathematics major or minor program must be fulfilled by Mathematics Department courses taken at Washington University. These courses are identified by the Department Code L24 that precedes the course name, and a course number that does not end in the letter "C" (a course number ending in "C" indicates that the course is a crosslisting of a course from some other university department). Courses transferred by students in Washington University approved Overseas Study Programs count as courses "taken at Washington University."

Courses from University College cannot be counted toward major requirements. Any exceptions must be made, in advance, by the Department's Undergraduate Committee. Approval requires a compelling special need for taking the course through University College and the Committee's judgment of the appropriateness of the course in a particular semester. If you ask for and receive approval for any such exception, please request and retain a printed or electronic copy of the approval to avoid any possible confusion when it's time for the Department to certify that major requirements have been fulfilled for graduation.

Graduate level courses in mathematics are numbered in the 500's. These courses are open to qualified undergraduates. The introductory graduate sequences (5021-22, 5031-32, 5041-42-43, 5051-52, 5061-62) are courses that some undergraduate majors might consider. Students who meet the prerequisites and are willing to put in the extra effort can find these courses very rewarding. But consult with your advisor before registering

An undergraduate with a plan for independent study and a faculty member to supervise the work can register for L24-400, Undergraduate Independent Study, using the section number corresponding to that faculty member. Section numbers are found under "Dept. Info" in the Course Listings online. Undergraduates should not register for L24-500 unless student and faculty member both agree that the independent study is clearly being done at the level of graduate work. At most 3 units of independent study or research work can count to fulfill major requirements. However, additional independent study credits not needed to meet major requirements are welcome, and credits for those do count toward the 120 credits needed to earn a bachelor's degree.

No upper level course used to satisfy a major requirement can also be counted to satisfy the requirements in any other major or minor: that is, no double-counting of courses between majors (or minors) is allowed This is policy established by the College, beginning with students who entered WU in Fall 2015 (the Class of 2019). It applies equally to second majors in mathematics by students from other schools such as Engineering of Business. The double counting policy does not apply to100-200 level courses.

When no double-counting is involved, a small number of approved course substitutions in each track are listed below. In all cases, at most one substitution can be made using a course not home-based in the Department of Mathematics. A course is home-based in Mathematics if it has department number L24 and the course number does not end with a "C" (a terminal "C" indicates that the course is home-based in another department and cross-listed as a mathematics course).

For example, the following courses are not home-based in the Mathematics Department: E35-317, ESE-319, E35-326,B59 QBA 120-121, L24-501C (home-based in the Physics Department), L24-440C (home-based in the Department ofComputer Science & Engineering), L24-403C (home-based in the Department of Philosophy).

When no double-counting is involved, majors may substitute at most one course for the purpose described in the list below. All other proposed substitutions or variations from this rule must be approved by the Department's Undergraduate Committee.

A course from another department that is cross-listed as an upper level L24 course: for example, L24-501C, L24-440C, or L24-403C. Such an L24 course always ends with a "C."

ESE 326 (Probability and Statistics for Engieering), as an upper level elective, or in place of Math 3200; but ESE 326 and Math 3200 cannot both be used to satisfy major requirements. Students who are doing a major following the probability/statistics track may be at some disadvantage from this substitution in more advanced topics courses which may assume a preliminary knowledge of SAS or R (introduced in 3200).

ESE 403 (Operations Research), as an upper level elective

ESE 411 (Numerical Methods), as an upper level elective or to replace Math 449. However, in this case, be sure to check with the instructor scheduled to teach Math 450 (a course whose topic can vary) to be sure the next offering of Math 450 won't assume some special material from Math 449. Math 449 and ESE 411 cannot both be used to satisfy major requirements.

ESE 319 (Engineering Mathematics B), as an upper level elective, but not as a substitute for Math 318 or 308. Students who took the former course ESE 317 (Engineering Mathematics), may substitute it in the same way as ESE 319. Note that ESE 318 (Engineering Mathematics A) is NOT an approved substitute in any track of the mathematics major.

Some Comments About Each Track

Track A: Traditional

The traditional track is probably the best choice for students who plan on "heavy duty" use of serious mathematics (for example, going to graduate school in mathematics or studying a heavily mathematical area of theoretical physics). Even if a student plans graduate work in statistics or "applied mathematics" a good graduate program will require a strong theoretical background. The traditional track is also a great choice for students who simply enjoy the rigor and beauty of more advanced mathematics.

Students planning on graduate school in mathematics should pick up the use of TeX, the premier software package for technical typesetting and word processing in mathematics. It is used to prepare manuscripts for nearly all professional papers and books. For example, see the Wikipedia entry for TeX. Some versions of TeX are available free online.

For students considering graduate study in mathematics, electives like Math 416 (Complex Variables), Math 4171 (Topology I), Math (Topology II) and Math 407 (Differential Geometry) are excellent choices, as well as at least one course in discrete mathematics (combinatorics, graph theory, or number theory).

Track B: Probability and Statistics

This track can provide useful general knowledge about statistics for students who will be looking for a job after the A.B. degree. In conjunction with some economics and finance courses, it provides an excellent background for entry into the actuarial profession. Students planning on graduate work in statistics should talk with their advisors about the possibility of also taking Math 4111-4121.

Students who plan to use statistics in practical settings are strongly encouraged to learn to use SAS. Familiarity with R may also be useful. SAS is thoroughly covered in Math 475, and several more advanced statistics courses make use of R.

Students with a sufficient background in computing may use Math 475 to fulfill the computer science requirement in this track. But in that case Math 475 cannot also count as one of the required upper level electives.

University College U20 4931-4941 may NOT be substituted for L24 Math 493-494, and U20-1201 cannot be substituted for the required computer science course.

Business School students who enrolled at WU as full-time students before Fall 2010 may substitute the pair of courses QBA 120 and 121 (Managerial Statistics I, II) for Math 3200. In that case, Math 3200 cannot also count toward the major. Students who began full time study at WU in Fall 2010 or later, cannot count QBA 120 and 121 as an upper level mathematics course to fulfill major requirements.

The Honors Program in Statistics consists of a very demanding set of courses in statistics and mathematics that gives an especially strong preparation for graduate school in statistics. Usually, a student needs to plan from the freshman year to work in all the courses. However, students can also announce their intention to follow this program later if they can still fit in all the coursework. Completion of the Honors Program in Statistics is noted in the "Honors" section of a student's permanent university record. Even if a student cannot complete all the courses in the Honors Program, it is a good guide for choosing courses in preparation for graduate studies in statistics.

Track C: Applied

The applied track can be done as a "stand-alone" major, but most students following this track combine it with substantial work in a math-related area such as physics, computer science, or engineering. With departmental approval, two substantially mathematical courses in a different department can sometimes be substituted for the physics or computer sciences requirement if the substitution forms part of a coherent "applications" program. To be "substantially mathematical," a course substitution from another department many need to be upper level.

Matlab is often used in Math 449. With a little basic computing background, you can pick it up in Math 449 as you go, but the whole experience would be more pleasant if you had some prior acquaintance with Matlab. Applied track majors might consider fulfilling part of the computing requirement early-on with CSE 200 which uses Matlab; or they might consider the 1-unit course CSE 100B as an extra course to ease into Matlab.

Computer Science Majors doing a second major in mathematics who receive a B or better in CSE 240, Logic and Discrete Mathematics, may replace Math 310 with an upper level math elective of their choice. (This would avoid some overlap in the course material, but the total number of upper level math credits required in the applied track would not be reduced). Conversely, the CSE Department may allow some majors to substitute Math 310 for their CSE 240 requirement (check with the CSE Department).

Students planning graduate work in applied mathematics might also discuss taking Math 4111 -4121 with their advisor.

Track D: Mathematics & Secondary Education

This track can be done only as a double major with secondary education. The courses required are typical of those required for certification as a secondary school mathematics teacher. Majors in this track should meet immediately with an advisor in the Education Department to understand its requirements and how to coordinate them with fulfilling the general education requirements of the College. The Education Department may also have good advice about certification requirements in other states and the possibilities for teaching in private schools that do not require state certification.

Some students who intend to become secondary teachers complete the requirements of some other math major track (often adding on Math 331 or 302 as extra electives) and plan to get teaching certification later in a post A.B. program. The Department of Education should be able to supply information about "certification later" programs.

Since this major track requires an additional major in secondary education (including student teaching), students pursuing this track who are uncertain about continuing with the education major should plan their math courses to make possible a later switch into one of the other tracks.

Also, placement by the Department of Education for student teaching requires a higher mathematics GPA than the math major itself (approximately a B average--check with the Department of Education). If it becomes clear to a student that such a math GPA looks unlikely, the student needs to consider a switch of major track.

Majors in this track need to finish the math major requirements by the end of the first semester of the senior year since they will be student teaching and taking professionally-oriented education courses during their eighth semester. Therefore some careful planning is needed: in particular, Math 302 is offered only in even-numbered fall semesters and Math 331 is offered only in odd-numbered fall semesters; and students should check with the Department of Computer Science and Engineering about its schedule for offering CSE 131.

With special permission from the department's Undergraduate Committee, students who complete a major in elementary or middle school education, including student teaching, may also complete a mathematics major using the "Secondary Education" track and subject to the same requirements. But such students should be aware that the course scheduling of Math 302 and 331 is planned to work together with the schedules (including student teaching) of students seeking secondary certification. There very well could be irreconcilable conflicts between math course requirements and required education courses or student teaching for elementary or middle school education majors.

Track E: Mathematics/Economics Emphasis

This track provides a strong mathematics background that resembles the traditional track, but with less algebra and more probability/statistics. It was designed after consulting with the Department of Economics and Olin Business School and should be particularly useful as a "second major" to economics majors planning to go to graduate school in economics or to students planning to enter a graduate program in finance. Both these options will make use of the required analysis sequence (4111-4121). The track could also be a good background, for a career in investment banking where pure math/economics candidates are sometimes attractive.

For students in the Business School only: MECO 290 (Microeconomics) may substitute for Economics 1011. MECO 292 (Global Economy) may be used in place of Economics 1021.